Medians and Altitudes — Coordinate Geometry Applications

9th Grade

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| By Thames

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| Questions: 20 | Updated: Dec 17, 2025

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1) The midpoint of side BC in ΔABC with B(2, 4) and C(6, 8) is

(3, 5)

(5, 7)

(2, 8)

(4, 6)

To find a midpoint, average the x- and y-coordinates of the endpoints: x = (2 + 6)/2 = 4, y = (4 + 8)/2 = 6. The midpoint (4, 6) represents the location exactly halfway along BC and is essential when constructing the median from the opposite vertex.

Explanation

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About This Quiz

Medians And Altitudes Coordinate Geometry Applications - Quiz

How do triangle medians and altitudes behave on the coordinate plane? In this quiz, you’ll apply coordinate geometry to locate midpoints, calculate slopes, and determine equations of key segments. You’ll practice finding points of intersection, interpreting geometric relationships algebraically, and verifying properties using precise computations. Step by step, you’ll see... see morehow coordinate tools bring clarity to triangle structure, helping you analyze medians and altitudes with confidence and accuracy.
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2) What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) For ΔABC with A(0, 0), B(6, 0), C(3, 6), the centroid coordinates are

(2, 3)

(3, 3)

(3, 2)

(4, 2)

The centroid (G) is found by averaging all vertex coordinates: x = (0 + 6 + 3)/3 = 3, y = (0 + 0 + 6)/3 = 2. Thus G(3, 2). The centroid’s coordinates correspond to the “center of mass,” balancing the triangle perfectly on that point.

Explanation

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3) The equation of the median from A(2, 4) to BC with B(6, 8), C(10, 4) is

Y − 4 = (x − 2)

Y − 4 = ⅓(x − 2)

Y = x + 1

Y − (−2) = 2(x − 0)

First find midpoint M of BC: M((6 + 10)/2, (8 + 4)/2) = (8, 6). Slope AM = (6 − 4)/(8 − 2) = 2/6 = ⅓. Equation → y − 4 = ⅓(x − 2). This represents the median line through A and the midpoint of BC.

Explanation

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4) In ΔABC with A(0, 0), B(6, 0), C(0, 8), find the length of the median from A to BC.

5 units

6 units

8 units

10 units

Midpoint M of BC = ((6 + 0)/2, (0 + 8)/2) = (3, 4). Length AM = √((3 − 0)² + (4 − 0)²) = √(9 + 16) = √25 = 5. The 3-4-5 result verifies a right-triangle relationship, confirming correctness.

Explanation

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5) The slope of the altitude to side BC in ΔABC with B(1, 2), C(5, 4) is

−2

1/2

−1/2

Slope of BC = (4 − 2)/(5 − 1) = 2/4 = 1/2. Altitudes are perpendicular, so the slope of the altitude = negative reciprocal = −2. Perpendicular slopes multiply to −1 (½ × −2 = −1).

Explanation

Slope of BC = (4 − 2)/(5 − 1) = 2/4 = 1/2. Altitudes are perpendicular, so the slope of the altitude = negative reciprocal = −2. Perpendicular slopes multiply to −1 (½ × −2 = −1).

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6) The coordinates of the centroid divide each median in the ratio

1:2

2:1

3:1

1:3

From vertex to centroid:centroid to midpoint = 2:1. This arises from the centroid’s definition G = ((x₁ + x₂ + x₃)/3,…). It physically means the centroid lies two-thirds of the way along every median from the vertex.

Explanation

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7) In ΔABC with A(2, 4), B(6, 8), C(10, 4), find the centroid.

(6, 5.33)

(5, 6)

(7, 5)

(4, 5)

x = (2 + 6 + 10)/3 = 18/3 = 6.

y = (4 + 8 + 4)/3 = 16/3 ≈ 5.33. Thus G(6, 5.33). This point lies inside the triangle and represents its geometric center.

Explanation

x = (2 + 6 + 10)/3 = 18/3 = 6.

y = (4 + 8 + 4)/3 = 16/3 ≈ 5.33. Thus G(6, 5.33). This point lies inside the triangle and represents its geometric center.

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8) If the altitude from A(0, 6) to BC (y = 2x + 4) is drawn, its slope is

−1/2

−2

1/2

Slope of BC = 2. Perpendicular lines have slopes that are negative reciprocals, so m_altitude = −1/2. Any line through A(0, 6) with this slope will be perpendicular to BC.

Explanation

Slope of BC = 2. Perpendicular lines have slopes that are negative reciprocals, so m_altitude = −1/2. Any line through A(0, 6) with this slope will be perpendicular to BC.

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9) The centroid of a triangle with vertices (−3, 4), (6, 1), (3, 7) is

(2, 5)

(3, 4)

(2, 4)

(1, 3)

x = (−3 + 6 + 3)/3 = 6/3 = 2; y = (4 + 1 + 7)/3 = 12/3 = 4. Hence G(2, 4). This calculation illustrates how coordinate averaging provides a quick method for verifying triangle centers.

Explanation

x = (−3 + 6 + 3)/3 = 6/3 = 2; y = (4 + 1 + 7)/3 = 12/3 = 4. Hence G(2, 4). This calculation illustrates how coordinate averaging provides a quick method for verifying triangle centers.

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10) Which statement about the centroid is true?

It divides the altitude in a 1:2 ratio.

It divides the median in a 2:1 ratio.

It lies outside a right triangle.

It bisects each angle.

The centroid lies along every median, exactly two-thirds of the distance from each vertex. It does not relate to altitudes or angle bisectors directly.

Explanation

The centroid lies along every median, exactly two-thirds of the distance from each vertex. It does not relate to altitudes or angle bisectors directly.

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11) The formula for the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is _____.

A centroid’s coordinates are the arithmetic mean of the vertex coordinates. This ensures equal “moment balance” along both x and y axes, acting as the triangle’s center of gravity.

Explanation

A centroid’s coordinates are the arithmetic mean of the vertex coordinates. This ensures equal “moment balance” along both x and y axes, acting as the triangle’s center of gravity.

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12) If the midpoint of BC is (3, 4), the equation of the median from A(0, 0) is _____.

Slope of AM = (4 − 0)/(3 − 0) = 4/3. Through origin → y = (4/3)x. This line connects vertex A to the midpoint M, defining the median in coordinate form.

Explanation

Slope of AM = (4 − 0)/(3 − 0) = 4/3. Through origin → y = (4/3)x. This line connects vertex A to the midpoint M, defining the median in coordinate form.

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13) The altitude from vertex A to side BC is always _____ to BC.

Altitudes are defined by perpendicularity. They form right angles with the opposite sides and are key in locating the orthocenter.

Explanation

Altitudes are defined by perpendicularity. They form right angles with the opposite sides and are key in locating the orthocenter.

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14) If the centroid is (5, 7) and the midpoint of the opposite side is (3, 4), the vertex coordinates are _____.

Use the section ratio 2:1 from vertex → centroid → midpoint. Vertex x = 3 + 2(5 − 3) = 7; y = 4 + 2(7 − 4) = 10. Hence (7, 10).

Explanation

Use the section ratio 2:1 from vertex → centroid → midpoint. Vertex x = 3 + 2(5 − 3) = 7; y = 4 + 2(7 − 4) = 10. Hence (7, 10).

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15) The equation of a line perpendicular to y = (1/2)x + 3 is _____.

Slope of given line = 1/2; perpendicular slope = −2. Replacing the slope yields y = −2x + k, where k is the y-intercept depending on the specific altitude’s position.

Explanation

Slope of given line = 1/2; perpendicular slope = −2. Replacing the slope yields y = −2x + k, where k is the y-intercept depending on the specific altitude’s position.

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16) The centroid can lie outside a triangle.

True

False

Because it’s the average of vertex coordinates, the centroid always falls inside the convex region bounded by the triangle.

Explanation

Because it’s the average of vertex coordinates, the centroid always falls inside the convex region bounded by the triangle.

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17) The orthocenter and centroid are always the same point.

True

False

They coincide only when the triangle is equilateral. In scalene or isosceles triangles these centers are distinct.

Explanation

They coincide only when the triangle is equilateral. In scalene or isosceles triangles these centers are distinct.

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18) If a median is drawn from a vertex, it divides the opposite side into two equal parts.

True

False

By definition, the median connects a vertex to the midpoint, ensuring the opposite side is bisected into equal segments.

Explanation

By definition, the median connects a vertex to the midpoint, ensuring the opposite side is bisected into equal segments.

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19) The altitude of a triangle always passes through its centroid.

True

False

The altitude passes through the orthocenter, not necessarily the centroid, except in equilateral triangles, where all centers coincide.

Explanation

The altitude passes through the orthocenter, not necessarily the centroid, except in equilateral triangles, where all centers coincide.

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20) The centroid divides each median in the ratio 2:1.

True

False

This ratio is consistent in all triangles, verified both geometrically and by coordinate analysis: the centroid is two-thirds along each median from the vertex.

Explanation

This ratio is consistent in all triangles, verified both geometrically and by coordinate analysis: the centroid is two-thirds along each median from the vertex.

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